We base our proof on a proof that the decision version of Hamiltonicity is polynomial
time computable in bounded clique-width graphs [1]}.
We prove that this algorithm can be parallelised and extended to the counting version.
Next, we show that for line graphs of bounded tree-width graphs which form the family of interest,
the clique-width expression can be inferred from the corresponding tree
decomposition. The tree decomposition itself is obtainable by the L-version of
Bodlaender's theorem [2]}.
